The entropy power inequality with quantum conditioning

The conditional entropy power inequality is a fundamental inequality in information theory, stating that the conditional entropy of the sum of two conditionally independent vector-valued random variables each with an assigned conditional entropy is minimum when the random variables are Gaussian. We prove the conditional entropy power inequality in the scenario where the conditioning system is quantum. The proof is based on the heat semigroup and on a generalization of the Stain inequality in the presence of quantum conditioning. The entropy power inequality with quantum conditioning will be a key tool of quantum information, with applications in distributed source coding protocols with the assistance of quantum entanglement.